# Find the measures of angles of pentagram?

• Feb 14th 2009, 12:42 PM
Rinnie
Find the measures of angles of pentagram?
What is the sum of the measures of angles A, B, C, D, and E?

http://i8.photobucket.com/albums/a32.../pentagram.jpg

I really need at least some kind of hint here, because I have no idea where to begin. No information is given for the picture.
• Feb 14th 2009, 01:18 PM
Plato
The sum of the angles in the pentagon, KJLMN, is $\displaystyle 3\pi =540^\circ$. WHY?
In $\displaystyle \Delta\text{ANK}$ the sum $\displaystyle \angle A + \angle 1 + \angle 2 = 180^ \circ$.
Moreover, by the external angle theorem $\displaystyle \angle K = \angle A + \angle 1$.
If you do this for each of the other four “external triangles” and all five you will see the answer.
• Feb 14th 2009, 01:56 PM
Rinnie
I did that before, and it got me no where. All I get is that:

$\displaystyle \angle N = \angle E + \angle 4$
$\displaystyle \angle M = \angle D + \angle 6$

etc.
• Feb 14th 2009, 02:05 PM
Plato
Quote:

Originally Posted by Rinnie
I did that before, and it got me no where.

You did not take it far enough then.
$\displaystyle \angle K + \angle N = 2\angle A + \angle 1 + \angle 2 = \angle A + 180^ \circ$.
Do that all around the "inside" pentagon.
Remember that sum of the angles in, KJLMN, is $\displaystyle 540^\circ$
• Feb 15th 2009, 10:49 AM
CaptainBlack
Quote:

Originally Posted by Rinnie
What is the sum of the measures of angles A, B, C, D, and E?

http://i8.photobucket.com/albums/a32.../pentagram.jpg

I really need at least some kind of hint here, because I have no idea where to begin. No information is given for the picture.

When you move a single vertex (at least if it remains on the same side of the line segment joining the oposite vertices) the angle sum remains invariant.

Since the given figure can be transformed by a sequence of such moves into a regular pentagram, the angle sum of the regular pentagram is equal to the angle sum of the given figure.

CB